Theorem bnj982 31148

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj982.1	⊢ (𝜑 → ∀𝑥𝜑)
bnj982.2	⊢ (𝜓 → ∀𝑥𝜓)
bnj982.3	⊢ (𝜒 → ∀𝑥𝜒)
bnj982.4	⊢ (𝜃 → ∀𝑥𝜃)

Assertion

Ref	Expression
bnj982	⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))

Proof of Theorem bnj982

Step	Hyp	Ref	Expression
1		df-bnj17 31054	. 2 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) ↔ ((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃))
2		bnj982.1	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
3		bnj982.2	. . . 4 ⊢ (𝜓 → ∀𝑥𝜓)
4		bnj982.3	. . . 4 ⊢ (𝜒 → ∀𝑥𝜒)
5	2, 3, 4	hb3an 2268	. . 3 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒))
6		bnj982.4	. . 3 ⊢ (𝜃 → ∀𝑥𝜃)
7	5, 6	hban 2267	. 2 ⊢ (((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃) → ∀𝑥((𝜑 ∧ 𝜓 ∧ 𝜒) ∧ 𝜃))
8	1, 7	hbxfrbi 1893	1 ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃) → ∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒 ∧ 𝜃))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072  ∀wal 1622   ∧ w-bnj17 31053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-10 2160  ax-12 2188

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-bnj17 31054

This theorem is referenced by:  bnj1096  31152  bnj1311  31391  bnj1445  31411